how does noise-to-signal ratio effect the data splitting in training,validation and test sets How will we divide our data set in training, validation and test set for model selection and model assessment? As in the book "Elements of statistical learning" page no.222 the author has mentioned that the number of observations in each of the three parts depends on noise-to-signal ratio and training sample size. then how will one decide that on the basis of noise-to-signal ratio that how much observations will be allocated to training set, validation and test set each? thanks in advance  
 A: You ask two questions:
First

how does noise-to-signal ratio affect the data splitting in
  training,validation and test sets?  

Second  

then how will one decide that on the basis of noise-to-signal ratio
  that how much observations will be allocated to training set,
  validation and test set each?

The first I'll answer as best I can and I'll explain why I can't give you an answer to the second one.
In the image below I have plotted a simple function $5x^2+0.3x$ which will serve as our signal - the true function we are trying to approximate with a model. The red circles represent the signal plus some added noise (top to bottom: $\mathcal{N}(0,10),\mathcal{N}(0,50),\mathcal{N}(0,100)$ ).
One can see that in the first plot we could still make out the signal fairly accurately based on the noisy signal, while it's harder with the second and unlikely in the third. 

Hastie et al. write about signal-to-noise in the section you're referring to in the context of how to segment one's data for model selection & assessment. Particularly, they write (emphasis is my own)

Again it is too difficult to give a general rule on how much
  training data is enough; among other things, this depends on the signal-to-noise ratio of the underlying function [...]

Indeed the question of 'how much training data is enough?' is one without general answer. Let's return to our example from above, but let's remove the pure signal line and only plot 20% of the data points.

Using only 20% of the data the top data will still give us a fit close enough to the signal, while with the second and third we're likely to get deviating results.
With little noise, it's possible to learn this well even with little data. But the noisier this signal gets, the more data we might need. Note, however, that in the third case merely "more data" alone wouldn't provide a better estimate, but it would probably also call for a more robust estimator in general.
Returning to the first question: it affects the number of observations we require to model the signal with sufficient precision, as I demonstrated. The more percent of the data we need for training, the less we can allocate to validation and test (assuming a split into disjunct sets and without drawing with replacement from the data).
The second question is more difficult. Unless you know what the signal is exactly, you'll only ever approximate the noise-to-signal ratio, with all the inaccuracies involved in it. Determining how large the proportion of your training set should be is not straightforward and requires exploration of the data and experimentation.
Matlab Code
rng(1)
x = linspace(-6,6,100);
signal = (5*x.^2+.3*x)';
noise1 = normrnd(0,10,length(x),1);
noise2 = normrnd(0,50,length(x),1);
noise3 = normrnd(0,100,length(x),1);
subplot (3,1,1)
plot(x,signal,'k-',x,signal+noise1,'ro')
ylim([-1,200])
legend({'signal','signal+noise'},'Location','north')
subplot (3,1,2)
plot(x,signal,'k-',x,signal+noise2,'ro')
ylim([-1,200])
subplot (3,1,3)
plot(x,signal,'k-',x,signal+noise3,'ro')
ylim([-1,200])

r=randi(100,20,1);
figure(2)
subplot (3,1,1)
plot(x(r),signal(r)+noise1(r),'ro')
ylim([-1,200])
subplot (3,1,2)
plot(x(r),signal(r)+noise2(r),'ro')
ylim([-1,200])
subplot (3,1,3)
plot(x(r),signal(r)+noise3(r),'ro')
ylim([-1,200])

